%I #51 Jun 25 2024 08:30:20
%S 5,13,17,37,53,61,73,89,97,109,113,137,149,157,173,193,197,233,257,
%T 269,277,293,313,317,337,353,373,389,397,421,433,457,557,577,593,613,
%U 617,653,661,673,677,701,733,757,761,773,797,821,829,853,857,877,937,953
%N Odd primes p with 4 zeros in any period of the Fibonacci numbers mod p.
%C Also, primes that do not divide any Lucas number. - _T. D. Noe_, Jul 25 2003
%C Although every prime divides some Fibonacci number, this is not true for the Lucas numbers. In fact, exactly 1/3 of all primes do not divide any Lucas number. See Lagarias and Moree for more details. The Lucas numbers separate the primes into three disjoint sets: (A053028) primes that do not divide any Lucas number, (A053027) primes that divide Lucas numbers of even index and (A053032) primes that divide Lucas numbers of odd index. - _T. D. Noe_, Jul 25 2003; revised by _N. J. A. Sloane_, Feb 21 2004
%C From _Jianing Song_, Jun 16 2024: (Start)
%C Primes p such that A001176(p) = 4.
%C For p > 2, p is in this sequence if and only if A001175(p) == 4 (mod 8), and if and only if A001177(p) is odd. For a proof of the equivalence between A001176(p) = 4 and A001177(p) being odd, see Section 2 of my link below.
%C This sequence contains all primes congruent to 13, 17 (mod 20). This corresponds to case (1) for k = 3 in the Conclusion of Section 1 of my link below. (End) [Comment rewritten by _Jianing Song_, Jun 16 2024 and Jun 25 2024]
%H T. D. Noe, <a href="/A053028/b053028.txt">Table of n, a(n) for n=1..1000</a>
%H C. Ballot and M. Elia, <a href="http://www.fq.math.ca/Papers1/45-1/quartballot01_2007.pdf">Rank and period of primes in the Fibonacci sequence; a trichotomy</a>, Fib. Quart., 45 (No. 1, 2007), 56-63 (The sequence B2).
%H Nicholas Bragman and Eric Rowland, <a href="https://arxiv.org/abs/2202.00704">Limiting density of the Fibonacci sequence modulo powers of p</a>, arXiv:2202.00704 [math.NT], 2022.
%H J. C. Lagarias, <a href="http://projecteuclid.org/euclid.pjm/1102706452">The set of primes dividing the Lucas numbers has density 2/3</a>, Pacific J. Math., 118. No. 2, (1985), 449-461.
%H J. C. Lagarias, <a href="http://projecteuclid.org/euclid.pjm/1102622818">Errata to: The set of primes dividing the Lucas numbers has density 2/3</a>, Pacific J. Math., 162, No. 2, (1994), 393-396.
%H Diego Marques and Pavel Trojovsky, <a href="https://doi.org/10.2478/tmmp-2014-0019">The order of appearance of the product of five consecutive Lucas numbers</a>, Tatra Mountains Math. Publ. 59 (2014), 65-77.
%H Pieter Moree, <a href="http://msp.org/pjm/1998/186-2/p03.xhtml">Counting Divisors of Lucas Numbers</a>, Pacific J. Math, Vol. 186, No. 2, 1998, pp. 267-284.
%H M. Renault, <a href="http://webspace.ship.edu/msrenault/fibonacci/fib.htm">Fibonacci sequence modulo m</a>
%H H. Sedaghat, <a href="https://www.fq.math.ca/Papers1/52-1/Sedaghat.pdf">Zero-Avoiding Solutions of the Fibonacci Recurrence Modulo A Prime</a>, Fibonacci Quart. 52 (2014), no. 1, 39-45. See p. 45.
%H Jianing Song, <a href="/A053027/a053027.pdf">Lucas sequences and entry point modulo p</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LucasNumber.html">Lucas Number</a>
%F A prime p = prime(i) is in this sequence if p > 2 and A001602(i) is odd. - _T. D. Noe_, Jul 25 2003
%t Lucas[n_] := Fibonacci[n+1] + Fibonacci[n-1]; badP={}; Do[p=Prime[n]; k=1; While[k<p&&Mod[Lucas[k], p]>0, k++ ]; If[k==p, AppendTo[badP, p]], {n, 200}]; badP
%Y Cf. A001175, A001177.
%Y Cf. A000204 (Lucas numbers), A001602 (index of the smallest Fibonacci number divisible by prime(n)).
%Y Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = m*x(n+1) + x(n). Let w(k) be the number of zeros in a fundamental period of {x(n)} modulo k.
%Y | m=1 | m=2 | m=3
%Y -----------------------------+----------+---------+---------
%Y The sequence {x(n)} | A000045 | A000129 | A006190
%Y The sequence {w(k)} | A001176 | A214027 | A322906
%Y Primes p such that w(p) = 1 | A112860* | A309580 | A309586
%Y Primes p such that w(p) = 2 | A053027 | A309581 | A309587
%Y Primes p such that w(p) = 4 | this seq | A261580 | A309588
%Y Numbers k such that w(k) = 1 | A053031 | A309583 | A309591
%Y Numbers k such that w(k) = 2 | A053030 | A309584 | A309592
%Y Numbers k such that w(k) = 4 | A053029 | A309585 | A309593
%Y * and also A053032 U {2}
%K nonn
%O 1,1
%A _Henry Bottomley_, Feb 23 2000
%E Edited: Name clarified. Moree and Renault link updated. Ballot and Elia reference linked. - _Wolfdieter Lang_, Jan 20 2015